Ab initio quantum chemistry methods

Accuracy and scaling

Ab initio electronic structure methods aim to calculate the many-electron function which is the solution of the non-relativistic electronic Schrödinger equation (in the Born–Oppenheimer approximation). The many-electron function is generally a linear combination of many simpler electron functions with the dominant function being the Hartree-Fock function. Each of these simple functions are then approximated using only one-electron functions. The one-electron functions are then expanded as a linear combination of a finite set of basis functions. This approach has the advantage that it can be made to converge to the exact solution, when the basis set tends toward the limit of a complete set and where all possible configurations are included (called "Full CI"). However this convergence to the limit is computationally very demanding and most calculations are far from the limit. Nevertheless important conclusions have been made from these more limited classifications.

One needs to consider the computational cost of ab initio methods when determining whether they are appropriate for the problem at hand. When compared to much less accurate approaches, such as molecular mechanics, ab initio methods often take larger amounts of computer time, memory, and disk space, though, with modern advances in computer science and technology such considerations are becoming less of an issue. The Hartree-Fock (HF) method scales nominally as N4 (N being a relative measure of the system size, not the number of basis functions) – e.g., if one doubles the number of electrons and the number of basis functions (double the system size), the calculation will take 16 (24) times as long per iteration. However, in practice it can scale closer to N3 as the program can identify zero and extremely small integrals and neglect them. Correlated calculations scale less favorably, though their accuracy is usually greater, which is the trade off one needs to consider. One popular method is Møller–Plesset perturbation theory (MP). To second order (MP2), MP scales as N4. To third order (MP3) MP scales as N6. To fourth order (MP4) MP scales as N7. Another method, coupled cluster with singles and doubles (CCSD), scales as N6 and extensions, CCSD(T) and CR-CC(2,3), scale as N6 with one noniterative step which scales as N7. Hybrid Density functional theory (DFT) methods using functionals which include Hartree–Fock exchange scale in a similar manner to Hartree–Fock but with a larger proportionality term and are thus more expensive than an equivalent Hartree–Fock calculation. Local DFT methods that do not include Hartree–Fock exchange can scale better than Hartree–Fock.

Linear scaling approaches

The problem of computational expense can be alleviated through simplification schemes.{{cite book

Classes of methods

The most popular classes of ab initio electronic structure methods:

Hartree–Fock methods

• (HF)
• (ROHF)
• (UHF)

Post-Hartree–Fock methods

• (MPn)
• (CI)
• (CC)
• (QCI)
• Sign learning kink-based (SiLK) quantum Monte Carlo

Multi-reference methods

• (MCSCF including CASSCF and RASSCF)
• (MRCI)
• (NEVPT)
• (CASPTn)
• (SUMR-CC)

Methods in detail

Hartree–Fock and post-Hartree–Fock methods

The simplest type of ab initio electronic structure calculation is the Hartree–Fock (HF) scheme, in which the instantaneous Coulombic electron-electron repulsion is not specifically taken into account. Only its average effect (mean field) is included in the calculation. This is a variational procedure; therefore, the obtained approximate energies, expressed in terms of the system's wave function, are always equal to or greater than the exact energy, and tend to a limiting value called the Hartree–Fock limit as the size of the basis is increased.{{cite book

;Example: Is the bonding situation in disilyne Si2H2 the same as in acetylene (C2H2)? A series of ab initio studies of Si2H2 is an example of how ab initio computational chemistry can predict new structures that are subsequently confirmed by experiment. They go back over 20 years, and most of the main conclusions were reached by 1995. The methods used were mostly post-Hartree–Fock, particularly configuration interaction (CI) and coupled cluster (CC). Initially the question was whether disilyne, Si2H2 had the same structure as ethyne (acetylene), C2H2. In early studies, by Binkley and Lischka and Kohler, it became clear that linear Si2H2 was a transition structure between two equivalent trans-bent structures and that the ground state was predicted to be a four-membered ring bent in a 'butterfly' structure with hydrogen atoms bridged between the two silicon atoms. Al2H2 and Ga2H2 have exactly the same isomers, in spite of having two electrons less than the Group 14 molecules. The only difference is that the four-membered ring ground state is planar and not bent. The cis-mono-bridged and vinylidene-like isomers are present. Experimental work on these molecules is not easy, but matrix isolation spectroscopy of the products of the reaction of hydrogen atoms and silicon and aluminium surfaces has found the ground state ring structures and the cis-mono-bridged structures for Si2H2 and Al2H2. Theoretical predictions of the vibrational frequencies were crucial in understanding the experimental observations of the spectra of a mixture of compounds. This may appear to be an obscure area of chemistry, but the differences between carbon and silicon chemistry is always a lively question, as are the differences between group 13 and group 14 (mainly the B and C differences). The silicon and germanium compounds were the subject of a Journal of Chemical Education article.

Valence bond methods

Valence bond (VB) methods are generally ab initio although some semi-empirical versions have been proposed. Current VB approaches are:

• (GVB)
• Modern valence bond theory (MVBT)

Quantum Monte Carlo methods

A method that avoids making the variational overestimation of HF in the first place is Quantum Monte Carlo (QMC), in its variational, diffusion, and Green's function forms. These methods work with an explicitly correlated wave function and evaluate integrals numerically using a Monte Carlo integration. Such calculations can be very time-consuming. The accuracy of QMC depends strongly on the initial guess of many-body wave-functions and the form of the many-body wave-function. One simple choice is Slater-Jastrow wave-function in which the local correlations are treated with the Jastrow factor.

Sign Learning Kink-based (SiLK) Quantum Monte Carlo (website): The Sign Learning Kink (SiLK) based Quantum Monte Carlo (QMC) method is based on Feynman's path integral formulation of quantum mechanics, and can reduce the minus sign problem when calculating energies in atomic and molecular systems.

References

References

1. Levine, Ira N.. (1991). "Quantum Chemistry". Prentice Hall.
2. Leach, Dr Andrew. (2001-01-30). "Molecular Modelling: Principles and Applications". Prentice Hall.
3. Friesner, Richard A.. (2005-05-10). "Ab initio quantum chemistry: Methodology and applications". Proceedings of the National Academy of Sciences of the United States of America.
4. Parr, Robert G.. "History of Quantum Chemistry".
5. Parr, R. G.. (1990). "On the genesis of a theory". Int. J. Quantum Chem..
6. Alireza Mashaghi et al., Hydration strongly affects the molecular and electronic structure of membrane phospholipids. J. Chem. Phys. 136, 114709 (2012) http://jcp.aip.org/resource/1/jcpsa6/v136/i11/p114709_s1 {{Webarchive. link. (2016-05-15)
7. Mischa Bonn et al., Interfacial Water Facilitates Energy Transfer by Inducing Extended Vibrations in Membrane Lipids, J Phys Chem, 2012 http://pubs.acs.org/doi/abs/10.1021/jp302478a
8. (5 January 2016). "Sign Learning Kink-based (SiLK) Quantum Monte Carlo for molecular systems". The Journal of Chemical Physics.